The Annals of Probability

Applications of Stein’s method for concentration inequalities

Sourav Chatterjee and Partha S. Dey

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Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie–Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős–Rényi random graph G(n, p) when p ≥ 0.31. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.

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Ann. Probab., Volume 38, Number 6 (2010), 2443-2485.

First available in Project Euclid: 24 September 2010

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60F10: Large deviations
Secondary: 60C05: Combinatorial probability 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Stein’s method Gibbs measures concentration inequality Ising model Curie–Weiss model large deviation Erdős–Rényi random graph exponential random graph


Chatterjee, Sourav; Dey, Partha S. Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 (2010), no. 6, 2443--2485. doi:10.1214/10-AOP542.

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